What is a good example of a nilpotent algebra?

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An example of a nilpotent algebra is mathfrak{t}_n(mathbb{C}), the complex-valued, strictly upper triangular matrices. The Lie bracket here is the standard commutator, [x,y]=xy-yx. Now clearly repeated applications of the commutator to upper triangular matrices will result in the matrix ‘level’ (the sub-diagonal above the main diagonal) decreasing, and thus the lower central series will terminate. section{Killing Form} The Killing form is a bilinear, symmetric form on mathfrak{g} defined as a function k: mathfrak{g} times mathfrak{g}...
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